Abstract: Rayleigh quotient iteration is an iterative algorithm for the calculation of approximate eigenvectors of a matrix. Given a matrix, the algorithm supplies a function whose iteration of an initial vector, , produces a sequence of vectors, . If the matrix is symmetric, then for almost any choice of the sequence will converge to an eigenvector at an eventually cubic rate. In this paper we show that there exist open sets of real matrices, each of which possesses an open set of initial vectors for which the algorithm will not converge to an eigenspace. The proof employs techniques from dynamical systems and bifurcation theory.